Local geometrized Rankin-Selberg method for GL(n)

Abstract

Following G. Laumon [12], to a nonramified $\ell$-adic local system $E$ of rank $n$ on a curve $X$ one associates a complex of $\ell$-adic sheaves $\sb n\mathscr {K}\sb E$ on the moduli stack of rank $n$ vector bundles on $X$ with a section, which is cuspidal and satisfies the Hecke property for $E$. This is a geometric counterpart of the well-known construction due to J. Shalika [19] and I. Piatetski-Shapiro [18]. We express the cohomology of the tensor product $\sb n\mathscr {K}\sb {E\sb 1}\otimes \sb n\mathscr {K}\sb {E\sb 2}$ in terms of cohomology of the symmetric powers of $X$. This may be considered as a geometric interpretation of the local part of the classical Rankin-Selberg method for ${\rm GL}(n)$ in the framework of the geometric Langlands program.